Let ℙkn denote n–dimensional projective space over k. The Zariski topology on ℙkn is defined to be the topology whose closed sets are the sets

V⁢(I):={x∈ℙkn∣f⁢(x)=0⁢ for all ⁢f∈I}⊂ℙkn,

where I⊂k⁢[X0,…,Xn] is any homogeneous ideal in the graded k–algebrak⁢[X0,…,Xn]. For any projective varietyV⊂ℙkn, the Zariski topology on V is defined to be the subspace topology induced on V as a subset of ℙkn.

The Zariski topology is the predominant topology used in the study of algebraic geometry. Every regular morphism of varieties is continuous in the Zariski topology (but not every continuous map in the Zariski topology is a regular morphism). In fact, the Zariski topology is the weakest topology on varieties making points in 𝔸k1 closed and regular morphisms continuous.